Lesson 2.2: Future Value (FV) & Compounding

Lesson Overview

The Future Value (FV) of money answers a practical question: If I invest money today, how much will I have later?

This lesson builds on Lesson 2.1’s TVM intuition and introduces the engine that powers long-term wealth: compounding. Compounding means you don’t just earn interest on your original deposit, you also earn interest on past interest. That “interest on interest” is why time is such a powerful ingredient in finance.

You’ll learn how to calculate future value, how compounding frequency changes results, and how to compare investments using the Effective Annual Rate (EAR).

Learning Objectives

Define Future Value (FV) and explain what it measures.
Calculate FV for a single lump sum with annual compounding.
Explain compounding frequency and compute FV with non-annual compounding.
Distinguish between APR/nominal rates and EAR.
Use EAR to compare savings or loan offers with different compounding periods.
Interpret FV results using timelines and units (years, months, periods).

Key Vocabulary

Principal (PV): the amount you start with (present value).
Future Value (FV): the amount you end with after growth over time.
Interest rate (r): the rate of return per period.
Number of periods (n): how many compounding periods occur.
Compounding: earning returns on both principal and accumulated returns.
APR (Nominal Rate): stated annual rate, not fully adjusted for compounding.
EAR (Effective Annual Rate): the true annual rate after compounding.

The Core Future Value Formula (Annual Compounding)

If you invest a lump sum today and it compounds once per year, the standard FV formula is:

FV = PV × (1 + r)ⁿ

PV = present value (starting amount)
r = interest rate per period (here, per year)
n = number of periods (years)

Read it like this: your money grows by a factor of (1 + r) each period, repeated n times.

Worked Example: A Simple FV Calculation

You deposit $2,000 into an account paying 5% per year for 3 years.

PV = 2,000
r = 0.05
n = 3

FV = 2,000 × (1.05)³ = 2,000 × 1.157625 = $2,315.25

Notice what happened: the interest earned in Year 1 becomes part of the base that earns interest in Years 2 and 3.

Compounding vs. Simple Interest

People sometimes confuse simple interest with compound interest. Simple interest only pays interest on the original principal. Compound interest pays interest on the principal and on accumulated interest.

If PV = $1,000, r = 10%, n = 3:

Simple interest: FV = 1,000 + (1,000 × 0.10 × 3) = $1,300
Compound interest: FV = 1,000 × (1.10)³ = 1,000 × 1.331 = $1,331

The longer the time horizon, the bigger the gap between simple and compound growth.

Compounding Frequency: Annual, Monthly, Daily…

Not all accounts compound once per year. Some compound monthly, daily, or even continuously. The more frequently interest is added to your balance, the faster the balance grows (assuming the same stated APR).

For compounding m times per year, the FV formula becomes:

FV = PV × (1 + r/m)^{m × t}

r = stated annual interest rate (APR)
m = compounding periods per year (12 for monthly, 365 for daily)
t = number of years

Worked Example: Monthly Compounding

You invest $5,000 at 6% APR compounded monthly for 2 years.

PV = 5,000
r = 0.06
m = 12
t = 2

FV = 5,000 × (1 + 0.06/12)^{12 × 2}
FV = 5,000 × (1.005)²⁴ ≈ 5,000 × 1.127 = $5,635 (approx.)

Your exact answer may differ slightly depending on rounding. In finance, keep extra decimals during intermediate steps.

APR vs. EAR: Why “6%” Might Not Mean 6%

Many rates you see advertised are APR (also called a nominal annual rate). APR tells you the stated annual rate, but it does not fully reflect the effect of compounding within the year.

The Effective Annual Rate (EAR) converts any compounding schedule into a single “true” annual rate, so you can compare apples-to-apples.

Effective Annual Rate (EAR) Formula

If an APR of r compounds m times per year:

EAR = (1 + r/m)^m - 1

This tells you: “If this account grows the way it actually compounds during the year, what annual rate does that equal?”

EAR Example: 12% APR Compounded Monthly

Suppose a credit card charges 12% APR compounded monthly.

r = 0.12
m = 12

EAR = (1 + 0.12/12)¹² - 1 = (1.01)¹² - 1 ≈ 1.1268 - 1 = 12.68%

Even though it says “12%,” the effective annual cost is higher because of monthly compounding.

Timeline Thinking: The #1 Habit That Prevents Mistakes

TVM errors usually come from mixing up periods. Before calculating FV, get clear on:

What is one period? (a year, a month, a day)
How many total periods are there?
Is the rate stated per period or per year?

A simple rule: your interest rate and your number of periods must match.
If you use monthly periods, use a monthly rate and count months.

Why Compounding Matters in Real Life

Savings & investing: retirement accounts, index funds, and reinvested dividends.
Debt: credit cards and loans can grow faster than people expect due to compounding.
Business decisions: reinvesting profits can create compounding growth in earnings and value.
Pricing: banks and lenders design products around compounding schedules.

Practice: Check Your Understanding

What does Future Value (FV) measure in one sentence?
Compute FV: If you invest $1,500 at 8% for 4 years (annual compounding), what is FV?
If an APR is compounded more frequently, does FV get bigger or smaller (holding APR constant)? Why?
What is the purpose of EAR?
A bank offers 5% APR compounded monthly and another offers 5% APR compounded annually. Which has the higher EAR?

Key Takeaways

FV tells you what today’s money becomes in the future.
Compounding means interest earns interest and growth accelerates over time.
Compounding frequency matters; more frequent compounding increases the effective growth rate.
EAR lets you compare rates fairly when compounding schedules differ.
Always align your rate per period with your number of periods.

What’s Next?

In Lesson 2.3: Present Value (PV) & Discounting, you’ll reverse the process: learn how to bring future money back to today, choose discount rates, and calculate present values.

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